Why do we lean towards left when the bus moves along a curved road to right.

Because your body has a natural instinctive response to resist external forces to stay upright and keep the vision-plane level. When you go around a corner in a moving bus, your body's inertia tends to keep it going straight in accord with the Newton's first law. However, since you are inside the turning bus, it doesn't feel to you as if you are just going straight, it feels like you are being pulled toward the outside wall of the bus as it goes around the turn. This apparent "force" is what you are working against when you lean "into" the turn.

According to the principle of equivalence gravity is indisdinguishable from acceleration. It means that inside a closed box you can not make an experiment that discerns gravity from acceleration.
Since the acceleration is to the right, we might just think of it as being attracted by a pseudo force of gravity to the left.

You can also think of it as the vehicle is pushing you to the right via friction (from the seat) but not directly pushing yout head, so your upper body is kind of being swung around by your point of contact.

You are reducing your velocity in your original direction as you transfer it to the new direction. When you lean to the left you are leaning in your original direction, same as when you press the brakes on your car and it pushes you forward.

The question is why do you feel a force when you accelerate. This is what you are doing when you are adding velocity to an orthogonal basis vector. The vehicle is changing it's relative velocity and so the seat grabs your bum and pulls you along with it. The fact that you feel a force in the opposite direction of acceleration agrees perfectly with the conservation of energy.